In a 5- to 6-quart pan over high heat, bring 2 1/2 to 3 quarts water to a boil. Meanwhile, if necessary, cut squid tubes crosswise into 1/4-inch-thick rings; leave tentacles whole. Rinse and drain squid or scallops.
Stir squid or scallops into boiling water; cover tightly and remove from heat. Let stand just until squid feels firm when pressed, about 1 minute (do not overcook or squid may turn rubbery), or scallops are barely opaque but still moist-looking in the center (cut to test), about 2 minutes. Drain, rinse in cold water, and drain again thoroughly. Pour the seafood into a large bowl.
Cut and discard peel and white membrane from oranges. One at a time, hold oranges over a bowl and cut between membranes and fruit to release segments into the bowl. Squeeze juice from membranes into the bowl; discard membranes.
Pour orange segments and juice into a wire strainer set over a 1- to 1 1/2-quart pan. Reserve segments. Bring juice to a boil over high heat and boil, stirring occasionally, until reduced to 2 tablespoons, 2 to 6 minutes. Remove from heat. Stir in oil, lime juice, shallots, ginger, sugar, and salt and pepper to taste.
Add orange dressing, orange segments, lettuce, jicama, and mint to squid; mix gently. Add more salt and pepper to taste. Divide mixture evenly among four dinner plates.